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System and method for generating pixel values for pixels in an image using strictly deterministic methodologies for generating sample points |
| 7133045 |
System and method for generating pixel values for pixels in an image using strictly deterministic methodologies for generating sample points
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| Patent Drawings: | |
| Inventor: |
Keller |
| Date Issued: |
November 7, 2006 |
| Application: |
09/884,861 |
| Filed: |
June 19, 2001 |
| Inventors: |
Keller; Alexander (Ulm, DE)
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| Assignee: |
Mental Images GmbH (Berlin, DE) |
| Primary Examiner: |
Chauhan; Ulka |
| Assistant Examiner: |
Repko; Jason M. |
| Attorney Or Agent: |
Jacobs & Kim LLPJacobs; David A. |
| U.S. Class: |
345/426 |
| Field Of Search: |
345/421; 345/426 |
| International Class: |
G06T 15/50 |
| U.S Patent Documents: |
6028606; 6529193 |
| Foreign Patent Documents: |
WO 98 59322 |
| Other References: |
Robert L. Cook, "Stochastic sampling in computer graphics," Jan. 1986, ACM Transactions on Graphics (TOG), vol. 5, Issue 1, p. 51-72. cited byexami- ner.
Tien-Tsin Wong, Wai-Shing Luk, Pheng-Ann Heng, "Sampling with Hammersley and Halton Points", 1997, Journal of Graphics Tools, vol. 2, No. 2, pp. 9-24. cited by examiner.
Andrew Keller, "Instant Radiosity," Aug. 3, 1997, Computer Graphics Proceedings, SIGGRAPH 97, p. 49-56. cited by examiner.
Robert L. Cook, "Stochastic Sampling in Computer Graphics," Jan. 1986, ACM Transactions on Graphics (TOG), vol. 5, Issue 1, p. 51-72. cited by exami- ner.
A. Keller, "Instant Radiosity," Computer Graphics Proceedings, SIGGRAPH 97, Los Angeles, Aug. 3-8, 1997 (Reading MA: Addison Wesley) pp. 49-56. cited by other.
IBM Technical Disclosure Bulletin "Quasi-Monte Carlo Rendering With Adaptive Sampling," vol. 39, No. 9, Sep. 1, 1996, pp. 215-224. cited by other.
K. Sung, et al., "Design and Implementation of the Maya Renderer," Computer Graphics And Applications, 1998, Pacific Graphics '98, Sixth Pacific Conf., Singapore, Oct. 26-29, 1998, IEEE Comp Soc Oct. 26, 1998, pp. 150-159, 231. cited byother. |
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| Abstract: |
A computer graphics system generates a pixel value for a pixel in an image, the pixel being representative of a point in a scene as recorded on an image plane of a simulated camera. The computer graphics system comprises a sample point generator and a function evaluator. The sample point generator is configured to generate a set of sample points representing at least one simulated element of the simulated camera, the sample points representing elements of, illustratively, for sample points on the image plane, during time interval during which the shutter is open, and on the lens, a Hammersley sequence, and, for use in global illumination, a scrambled Halton sequence. The function evaluator configured to generate at least one value representing an evaluation of said selected function at one of the sample points generated by said sample point generator, the value generated by the function evaluator corresponding to the pixel value. |
| Claim: |
What is claimed as new and desired to be secured by Letters Patent of the United States is:
1. A computer graphics system for generating a pixel value for a pixel in an image, the pixel beingrenresentative of a point in a scene as recorded on an image plane of a simulated camera, the computer aranhics system compnsmg: A. a sample point generator configured to generate a set of sample points representing at least one simulated element of thesimulated camera, the sample points representing elements of a Hammersley sequence; and B. a function evaluator configured to generate at least one value representing an evaluation of said selected function at one of the sample points generated by saidsample point generator, the value generated by the function evaluator corresponding to the pixel value; in which the sample point generator is configured to generate sample position points x.sub.i. representing jittered sample point positions on asubpixel grid for at least one pixel on the image plane for use by the function evaluator in evaluating the selected function; and in which the sample point generator is configured to generate the sample position points x in accordance with.PHI..function..PHI..function..sigma..function..sigma..function. ##EQU00034## where (s.sub.x,s.sub.y) are subpixel coordinates mapped onto strata coordinates (j,k):=(s.sub.x mod2.sup.n, s.sub.y mod2.sup.n), and instance number "i" corresponds toi=j2.sup.n=.sigma.(k) where integer permutation .sigma.(k):=2.sup.n.PHI..sub.2(k) for 0.ltoreq.k<2.sup.n for "n" a selected integer, and .PHI..sub.b(q) is the radical inverse function of "q" in base "b."
2. A computer graphics system for generating a pixel value for a pixel in an image, the pixel being representative of a point in a scene as recorded on an image plane of a simulated camera, the computer graphics system comprising: A. a samplepoint generator configured to generate a set of sample points representing at least one simulated element of the simulated camera, the sample points representing elements of a Hammersley sequence; and B. a function evaluator configured to generate atleast one value representing an evaluation of said selected function at one of the sample points generated by said sample point generator, the value generated by the function evaluator corresponding to the pixel value; in which the sample pointgenerator is configured to generate sample position points x.sub.i representing jittered sample point positions on a subpixel grid for at least one pixel on the image plane for use by the function evaluator in evaluating the selected function; in whichthe simulated camera is to be provided with a shutter, and the sample point generator is configured to generate, for at least one sample point position on a subpixel for at least one pixel on the image plane, sample time points t.sub.i,j represnting "j"points in time during a time interval t.sub.0 to t.sub.0=T during which the shutter is to be open for use by the function evaluator in evaluating the selected function; and in which the sample point generator is configured to generate the sample timepoints t.sub.i,j in accordance with .PHI..function..sym. ##EQU00035## where N.sub.T corresponds to a predetermined number of times at which sample time points t.sub.i,j are to be generated and .PHI.(q) is the radical inverse function of "q" in base "b."
3. A computer graphics system for generating a pixel value for a pixel in an image, the pixel being representative of a point in a scene as recorded on an image plane of a simulated camera, the computer graphics system comprising: A. a samplepoint generator configured to generate a set of sample points representing at least one simulated element of the simulated camera, the sample points representing elements of a Hammersley sequence; and B. a function evaluator configured to generate atleast one value representing an evaluation of said selected function at one of the sample points generated by said sample point generator, the value generated by the function evaluator corresponding to the pixel value; in which the sample pointgenerator is configured to generate sample position points x.sub.i representing jittered sample point positions on a subpixel grid for at least one pixel on the image plane for use by the function evaluator in evaluating the selected function; in whichthe simulated camera is to be provided with a shutter, and the sample point generator is configured to generate, for at least one sample point position on a subpixel for at least one pixel on the image plane, sample time points t.sub.i,j representing "j"points in time during a time interval t.sub.0 to t.sub.0=T during which the shutter is to be open for use by the function evaluator in evaluating the selected function; in which the simulated camera is to be provided with a lens, and the sample pointgenerator is configured to generate, for at least one sample point position and at least one sample time point, sample lens position points y.sub.i,j,k representing "k" points on the lens for use by the function evaluator in evaluating the selectedfunction; and in which the sample point generator is configured to generate the sample lens points in accordance with .PHI..function..sigma..sym..PHI..function..sigma..sym..PHI..function. ##EQU00036## where .PHI..sub.b(q,.sigma.) is the scrambledradical inverse function of "q" in base "b" and ".sym." refers to addition modulo a predetermined value and instance number "i" corresponds to i=j2.sup.n=.sigma.(k) where integer permutation .sigma.(k):=2.sup.n.PHI..sub.2(k) for 0.ltoreq.k<2.sup.n for"n" a selected integer.
4. A computer graphics system as defined in claim 3 in which the predetermined value is "one."
5. A computer graphics method for generating a pixel value for a pixel in an image, the pixel being representative of a point in a scene as recorded on an image plane of a simulated camera, the computer graphics method comprising: A. a samplepoint generating step in which a set of sample points are generated representing at least one simulated element of the simulated camera, the sample points representing elements of a Hammersley sequence; and B. a function evaluation step in which atleast one value is generated representing an evaluation of said selected function at one of the sample points generated by said sample point generator, the value generated by the function evaluator corresponding to the pixel value; in which the samplepoint generating step includes the step of generating sample position points x.sub.i representing jittered sample point positions on a subpixel grid for at least one pixel on the image plane for use by the function evaluator in evaluating the selectedfunction; and in which the sample point generating step includes the step of generating the sample position points x.sub.i in accordance with .PHI..function..PHI..function..sigma..function..sigma..function. ##EQU00037## where (s.sub.x,s.sub.y) aresubpixel coordinates mapped onto strata coordinates (j,k):=(s.sub.x mod2.sup.n,s2.sup.n), and instance number "i" corresponds to i=j2.sup.n=.sigma.(k) where integer permutation .sigma.(k):=2.sup.n.PHI..sub.2(k) for 0.ltoreq.k<2.sup.n for "n" aselected integer, and .PHI..sub.b(q) is the radical inverse function of "q" in base "b" generating a display controlling electronic output from the pixel value.
6. A computer graphics method for generating a pixel value for a pixel in an image, the pixel being representative of a point in a scene as recorded on an image plane of a simulated camera, the computer graphics method comprising: A. a samplepoint generating step in which a set of sample points are generated representing at least one simulated element of the simulated camera, the sample points representing elements of a Hammersley sequence; and B. a function evaluation step in which atleast one value is generated representing an evaluation of said selected function at one of the sample points generated by said sample point generator, the value generated by the function evaluator corresponding to the pixel value; in which the samplepoint generating step includes the step of generating sample position points x.sub.i representing jittered sample point positions on a subpixel grid for at least one pixel on the image plane for use by the function evaluator in evaluating the selectedfunction; in which the simulated camera is to be provided with a shutter, and the sample point generating step includes the step of generating, for at least one sample point position on a subpixel for at least one pixel on the image plane, sample timepoints t.sub.i,j representing "i" points in time during a time interval t.sub.0 to t.sub.0=T during which the shutter is to be open for use by the function evaluator in evaluating the selected function; and in which the sample point generating stepincludes the step of generating the sample time points t.sub.i,j in accordance with .PHI..function..sym. ##EQU00038## where N.sub.T corresponds to a predetermined number of times at which sample time points t.sub.i,j are to be generated and.PHI..sub.b(q) is the radical inverse function of "q" in base "b" generating a display controlling electronic output from the pixel value.
7. A computer graphics method as defined in claim 6 in which the simulated camera is to be provided with a lens, and the sample point generating step includes the step of generating, for at least one sample point position and at least onesample time point, sample lens position points Y.sub.i,j,k representing "k" points on the lens for use by the function evaluator in evaluating the selected function.
8. A computer graphics method as defined in claim 1 in which the sample point generating step includes the step of generating the sample lens points in accordance with .PHI..function..sigma..sym..PHI..function..sigma..sym..PHI..function. ##EQU00039## where .PHI..sub.b(q,.sigma.) is the scrambled radical inverse function of "q" in base "b" and ".sym." refers to addition modulo a predetermined value and instance number "i" corresponds to i=j2.sup.n+.sigma.(k) where integer permutation.sigma.(k):=2.sup.n.PHI..sub.2(k) for 0.ltoreq.k<2.sup.n for "n" a selected integer.
9. A computer graphics method as defined in claim 8 in which the predetermined value is "one." |
| Description: |
INCORPORATION BY REFERENCE
U.S. patent application Ser. No. 08/880,418, filed Jun. 23, 1997, in the names of Martin Grabenstein, et al., entitled "System And Method For Generating Pixel Values For Pixels In An Image Using Strictly Deterministic Methodologies ForGenerating Sample Points," (hereinafter referred to as the Grabenstein application) assigned to the assignee of this application, incorporated by reference.
FIELD OF THE INVENTION
The invention relates generally to the field of computer graphics, and more particularly to systems and methods for generating pixel values for pixels in the image.
BACKGROUND OF THE INVENTION
In computer graphics, a computer is used to generate digital data that represents the projection of surfaces of objects in, for example, a three-dimensional scene, illuminated by one or more light sources, onto a two-dimensional image plane, tosimulate the recording of the scene by, for example, a camera. The camera may include a lens for projecting the image of the scene onto the image plane, or it may comprise a pinhole camera in which case no lens is used. The two-dimensional image is inthe form of an array of picture elements (which are variable termed "pixels" or "pels"), and the digital data generated for each pixel represents the color and luminance of the scene as projected onto the image plane at the point of the respective pixelin the image plane. The surfaces of the objects may have any of a number of characteristics, including shape, color, specularity, texture, and so forth, which are preferably rendered in the image as closely as possible, to provide a realistic-lookingimage.
Generally, the contributions of the light reflected from the various points in the scene to the pixel value representing the color and intensity of a particular pixel are expressed in the form of the one or more integrals of relativelycomplicated functions. Since the integrals used in computer graphics generally will not have a closed-form solution, numerical methods must be used to evaluate them and thereby generate the pixel value. Typically, a conventional "Monte Carlo" methodhas been used in computer graphics to numerically evaluate the integrals. Generally, in the Monte Carlo method, to evaluate an integral
.intg..times..function..times..times.d ##EQU00001## where f(x) is a real function on the real numerical interval from zero to 1, inclusive, first a number "N" statistically-independent random numbers xi, i=1, . . . , N, are generated over theinterval. The random numbers xi are used as sample points for which sample values f(xi) are generated for the function f(x), and an estimate {overscore (f)} for the integral is generated as
.apprxeq..times..times..times..function. ##EQU00002## As the number of random numbers used in generating the sample points f(xi) increases, the value of the estimate {overscore (f)} will converge toward the actual value of the integral<f>. Generally, the distribution of estimate values that will be generated for various values of "N," that is, for various numbers of samples, of being normal distributed around the actual value with a standard deviation .sigma. which can beestimated by
.sigma..times. ##EQU00003## if the values x.sub.i used to generate the sample values f(x.sub.i) are statistically independent, that is, if the values x.sub.i are truly generated at random.
Generally, it has been believed that random methodologies like the Monte Carlo method are necessary to ensure that undesirable artifacts, such as Moire patterns and aliasing and the like, which are not in the scene, will not be generated in thegenerated image. However, several problems arise from use of the Monte Carlo method in computer graphics. First, since the sample points x.sub.i used in the Monte Carlo method are randomly distributed, they may clump in various regions over theinterval over which the integral is to be evaluated. Accordingly, depending on the set of random numbers which are generated, in the Monte Carlo method for significant portions of the interval there may be no sample points x.sub.i for which samplevalues f(x.sub.i) are generated. In that case, the error can become quite large. In the context of generating a pixel value in computer graphics, the pixel value that is actually generated using the Monte Carlo method may not reflect some elementswhich might otherwise be reflected if the sample points x.sub.i were guaranteed to be more evenly distributed over the interval. This problem can be alleviated somewhat by dividing the interval into a plurality of sub-intervals, but it is generallydifficult to determine a priori the number of sub-intervals into which the interval should be divided, and, in addition, in a multi-dimensional integration region (which would actually be used in computer graphics, instead of the one-dimensional intervaldescribed here) the partitioning of the region into sub-regions of equal size can be quite complicated.
In addition, since the method makes use of random numbers, the error |{overscore (f)}-<f>| (where |x| represents the absolute value of the value "x") between the estimate value {overscore (f)} and actual value <f> is probabilistic,and, since the error values for various large values of "N" are close to normal distribution around the actual value <f>, only sixty-eight percent of the estimate values {overscore (f)} that might be generated are guaranteed to lie within onestandard deviation of the actual value <f>.
Furthermore, as is clear from equation (3), the standard deviation a decreases with increasing numbers of samples N, proportional to the reciprocal of square root of "N" (that is, 1/ {square root over (N)}). Thus, if it is desired to reduce thestatistical error by a factor of two, it will be necessary to increase the number of samples N by a factor of four, which, in turn, increases the computational load that is required to generate the pixel values, for each of the numerous pixels in theimage.
Additionally, since the Monte Carlo method requires random numbers, an efficient mechanism for generating random numbers is needed. Generally, digital computers are provided with so-called "random number" generators, which are computer programswhich can be processed to generate a set of numbers that are approximately random. Since the random number generators use deterministic techniques, the numbers that are generated are not truly random. However, the property that subsequent randomnumbers from a random number generator are statistically independent should be maintained by deterministic implementations of pseudo-random numbers on a computer.
The Grabenstein application describes a computer graphics system and method for generating pixel values for pixels in an image using a strictly deterministic methodology for generating sample points, which avoids the above-described problems withthe Monte Carlo method. The strictly deterministic methodology described in the Grabenstein application provides a low-discrepancy sample point sequence which ensures, a priori, that the sample points are generally more evenly distributed throughout theregion over which the respective integrals are being evaluated. In one embodiment, the sample points that are used are based on a so-called Halton sequence. See, for example, J. H. Halton, Numerische Mathematik, Vol. 2, pp. 84 90 (1960) and W. H.Press, et al., Numerical Recipes in Fortran (2d Edition) page 300 (Cambridge University Press, 1992). In a Halton sequence generated for number base "p," where base "p" is a selected prime number, the "k-th" value of the sequence, represented byH.sub.p.sup.k is generated by use of a "radical inverse" operation, that is, by (1) writing the value "k" as a numerical representation of the value in the selected base "p," thereby to provide a representation for the value as D.sub.MD.sub.M-1 . . .D.sub.2D.sub.1, where D.sub.m (m=1, 2, . . . , M) are the digits of the representation, (2) putting a radix point (corresponding to a decimal point for numbers written in base ten) at the least significant end of the representation D.sub.MD.sub.M-1 . .. D.sub.2D.sub.1 written in step (1) above, and (3) reflecting the digits around the radix point to provide 0.D.sub.1D.sub.2 . . . D.sub.M-1D.sub.M, which corresponds to H.sub.p.sup.k. It will be appreciated that, regardless of the base "p" selectedfor the representation, for any series of values, one, two, . . . "k," written in base "p," the least significant digits of the representation will change at a faster rate than the most significant digits. As a result, in the Halton sequenceH.sub.p.sup.1, H.sub.p.sup.2, . . . H.sub.p.sup.k, the most significant digits will change at the faster rate, so that the early values in the sequence will be generally widely distributed over the interval from zero to one, and later values in thesequence will fill in interstices among the earlier values in the sequence. Unlike the random or pseudo-random numbers used in the Monte Carlo method as described above, the values of the Halton sequence are not statistically independent; on thecontrary, the values of the Halton sequence are strictly deterministic, "maximally avoiding" each other over the interval, and so they will not clump, whereas the random or pseudo-random numbers used in the Monte Carlo method may clump.
It will be appreciated that the Halton sequence as described above provides a sequence of values over the interval from zero to one, inclusive along a single dimension. A multi-dimensional Halton sequence can be generated in a similar manner,but using a different base for each dimension.
A generalized Halton sequence, of which the Halton sequence described above is a special case, is generated as follows. For each starting point along the numerical interval from zero to one, inclusive, a different Halton sequence is generated. Defining the pseudo-sum x.sym..sub.py for any x and y over the interval from zero to one, inclusive, for any integer "p" having a value greater than two, the pseudo-sum is formed by adding the digits representing "x" and "y" in reverse order, from themost-significant digit to the least-significant digit, and for each addition also adding in the carry generated from the sum of next more significant digits. Thus, if "x" in base "p" is represented by 0.X.sub.1X.sub.2 . . . X.sub.M-1X.sub.M, where each"X.sub.m" is a digit in base "p," and if "y" in base "p" is represented by 0.Y.sub.1Y.sub.2 . . . Y.sub.N-1Y.sub.N, where each "Y.sub.n" is a digit in base "p" (and where "M," the number of digits in the representation of "x" in base "p", and "N," thenumber of digits in the representation of "y" in base "p", may differ), then the pseudo-sum "z" is represented by 0.Z.sub.1Z.sub.2 . . . Z.sub.L-1Z.sub.L, where each "Z.sub.1" is a digit in base "p" given by Z.sub.l=(X.sub.l+Y.sub.l+C.sub.l) mod p,where "mod" represents the modulo function, and
.times..times..gtoreq. ##EQU00004## is a carry value from the "l-1st" digit position, with C.sub.l being set to zero.
Using the pseudo-sum function as described above, the generalized Halton sequence that is used in the system described in the Grabenstein application is generated as follows. If "p" is an integer, and x.sub.0 is an arbitrary value on theinterval from zero to one, inclusive, then the "p"-adic von Neumann-Kakutani transformation T.sub.p(x) is given by
.function..times..times..sym..times. ##EQU00005## and the generalized Halton sequence x.sub.0, x.sub.1, x.sub.2 . . . is defined recursively as x.sub.n+1=T.sub.p(x.sub.n) (5) From equations (4) and (5), it is clear that, for any value for "p,"the generalized Halton sequence can provide that a different sequence will be generated for each starting value of "x," that is, for each x.sub.0. It will be appreciated that the Halton sequence H.sub.p.sup.k as described above is a special case of thegeneralized Halton sequence (equations (4) and (5)) for x.sub.0=0.
The use of a strictly deterministic low-discrepancy sequence can provide a number of advantages over the random or pseudo-random numbers that are used in connection with the Monte Carlo technique. Unlike the random numbers used in connectionwith the Monte Carlo technique, the low discrepancy sequences ensure that the sample points are more evenly distributed over a respective region or time interval, thereby reducing error in the image which can result from clumping of such sample pointswhich can occur in the Monte Carlo technique. That can facilitate the generation of images of improved quality when using the same number of sample points at the same computational cost as in the Monte Carlo technique.
SUMMARY OF THE INVENTION
The invention provides a new and improved computer graphics system and method for generating pixel values for pixels in the image using a strictly deterministic methodology for generating sample points for use in evaluating integrals definingaspects of the image.
In brief summary, the invention provides, in one aspect, a computer graphics system for generating a pixel value for a pixel in an image, the pixel being representative of a point in a scene as recorded on an image plane of a simulated camera. The computer graphics system comprises a sample point generator and a function evaluator. The sample point generator is configured to generate a set of sample points representing at least one simulated element of the simulated camera, the sample pointsrepresenting elements of, illustratively, for sample points on the image plane, during time interval during which the shutter is open, and on the lens, a Hammersley sequence, and, for use in global illumination, a scrambled Halton sequence. The functionevaluator configured to generate at least one value representing an evaluation of said selected function at one of the sample points generated by said sample point generator, the value generated by the function evaluator corresponding to the pixel value.
BRIEF DESCRIPTION OF THE DRAWINGS
This invention is pointed out with particularity in the appended claims. The above and further advantages of this invention may be better understood by referring to the following description taken in conjunction with the accompanying drawings,in which:
FIG. 1 depicts an illustrative computer graphics system constructed in accordance with the invention
FIG. 2 is a flowchart illustrating an overall method in accord with the invention.
DETAILED DESCRIPTION OF AN ILLUSTRATIVE EMBODIMENT
The invention provides an computer graphic system and method for generating pixel values for pixels in an image of a scene, which makes use of a strictly-deterministic methodology for generating sample points for use in generating sample valuesfor evaluating the integral or integrals whose function(s) represent the contributions of the light reflected from the various points in the scene to the respective pixel value, rather than the random or pseudo-random Monte Carlo methodology which hasbeen used in the past. The strictly-deterministic methodology ensures a priori that the sample points will be generally more evenly distributed over the interval or region over which the integral(s) is (are) to be evaluated in a low-discrepancy manner.
FIG. 1 attached hereto depicts an illustrative computer system 10 that makes use of such a strictly deterministic methodology. With reference to FIG. 1, the computer system 10 in one embodiment includes a processor module 11 and operatorinterface elements comprising operator input components such as a keyboard 12A and/or a mouse 12B (generally identified as operator input element(s) 12) and an operator output element such as a video display device 13. The illustrative computer system10 is of the conventional stored-program computer architecture. The processor module 11 includes, for example, one or more processor, memory and mass storage devices, such as disk and/or tape storage elements (not separately shown), which performprocessing and storage operations in connection with digital data provided thereto. The operator input element(s) 12 are provided to permit an operator to input information for processing. The video display device 13 is provided to display outputinformation generated by the processor module 11 on a screen 14 to the operator, including data that the operator may input for processing, information that the operator may input to control processing, as well as information generated during processing. The processor module 11 generates information for display by the video display device 13 using a so-called "graphical user interface" ("GUI"), in which information for various applications programs is displayed using various "windows." Although thecomputer system 10 is shown as comprising particular components, such as the keyboard 12A and mouse 12B for receiving input information from an operator, and a video display device 13 for displaying output information to the operator, it will beappreciated that the computer system 10 may include a variety of components in addition to or instead of those depicted in FIG. 1.
In addition, the processor module 11 includes one or more network ports, generally identified by reference numeral 14, which are connected to communication links which connect the computer system 10 in a computer network. The network portsenable the computer system 10 to transmit information to, and receive information from, other computer systems and other devices in the network. In a typical network organized according to, for example, the client-server paradigm, certain computersystems in the network are designated as servers, which store data and programs (generally, "information") for processing by the other, client computer systems, thereby to enable the client computer systems to conveniently share the information. Aclient computer system which needs access to information maintained by a particular server will enable the server to download the information to it over the network. After processing the data, the client computer system may also return the processeddata to the server for storage. In addition to computer systems (including the above-described servers and clients), a network may also include, for example, printers and facsimile devices, digital audio or video storage and distribution devices, andthe like, which may be shared among the various computer systems connected in the network. The communication links interconnecting the computer systems in the network may, as is conventional, comprise any convenient information-carrying medium,including wires, optical fibers or other media for carrying signals among the computer systems. Computer systems transfer information over the network by means of messages transferred over the communication links, with each message including informationand an identifier identifying the device to receive the message.
It will be helpful to initially provide some background on operations performed by the computer graphics system in generating an image. Generally, the computer graphic system generates an image attempts to simulate an image of a scene that wouldbe generated by a camera. The camera includes a shutter that will be open for a predetermined time T starting at a time t.sub.0 to allow light from the scene to be directed to an image plane. The camera may also include a lens that serves to focuslight from the scene onto the image plane. The average radiance flux L.sub.m,n through a pixel at position (m,n) on an image plane P, which represents the plane of the camera's recording medium, is determined by
.times..intg..times..intg..times..intg..times..function..function..omega..- function..times..function..times..times.d.times..times.d.times..times.d ##EQU00006## where "A.sub.P" refers to the area of the pixel, A.sub.L refers to the area of theportion of the lens through which rays of light pass from the scene to the pixel, and f.sub.m,n represents a filtering kernel associated with the pixel. An examination of the integral in equation (6) will reveal that the variables of integration, "x,""y" and "t," are essentially dummy variables with the variable "y" referring to integration over the lens area (A.sub.L), the variable "t" referring to integration over time (the time interval from t.sub.0 to t.sub.0+T) and the variable "x" referring tointegration over the pixel area (A.sub.P).
The value of the integral in equation (6) is approximated by identifying N.sub.P sample points x.sub.i in the pixel area, and, for each sample point, shooting N.sub.T rays at times t.sub.i,j in the time interval t.sub.0 to t.sub.0+T through thefocus into the scene, with each ray spanning N.sub.L sample points y.sub.i,j,k on the lens area A.sub.L. The manner in which subpixel jitter positions x.sub.i, points in time t.sub.i,j and positions on the lens y.sub.i,j,k are determined will bedescribed below. These three parameters determine the primary ray hitting the scene geometry in h(x.sub.i,t.sub.i,j,y.sub.i,j,k) with the ray direction .omega.(x.sub.i,t.sub.i,j,y.sub.i,j,k). In this manner, the value of the integral in equation (6)can be approximated as
.apprxeq..times..times..times..times..times..times..times..times..times..f- unction..function..omega..function..times..function. ##EQU00007## where "N" is the total number of rays directed at the pixel.
It will be appreciated that rays directed from the scene toward the image can comprise rays directly from one or more light sources in the scene, as well as rays reflected off surfaces of objects in the scene. In addition, it will be appreciatedthat a ray that is reflected off a surface may have been directed to the surface directly from a light source, or a ray that was reflected off another surface. For a surface that reflects light rays, a reflection operator T.sub.fr is defined thatincludes a diffuse portion T.sub.fd, a glossy portion T.sub.fg and a specular portion T.sub.fs, or T.sub.fr=T.sub.fd+T.sub.fg+T.sub.fs (8). In that case, the Fredholm integral equation L=L.sub.e+T.sub.frL governing light transport can be represented asL=L.sub.eT.sub.fr-fsL.sub.e+T.sub.fs(L-L.sub.e)+T.sub.fsL+T.sub.fdT.sub.f- g+fsL+T.sub.fdT.sub.fdL (9), where transparency has been ignored for the sake of simplicity; transparency is treated in an analogous manner. The individual terms in equation (9)are
(i) L.sub.e represents flux due to a light source;
(ii) T.sub.fr-fsL.sub.e (where T.sub.fr-fs=T.sub.fr-T.sub.fs) represents direct illumination, that is, flux reflected off a surface that was provided thereto directly by a light source; the specular component, associated with the specular portionT.sub.fs of the reflection operator, will be treated separately since it is modeled as a .delta.-distribution;
(iii) T.sub.fg(L-L.sub.e) represents glossy illumination, which is handled by recursive distribution ray tracing, where, in the recursion, the source illumination has already been accounted for by the direct illumination (item (ii) above);
(iv) T.sub.fsL represents a specular component, which is handled by recursively using "L" for the reflected ray;
(v) T.sub.fdT.sub.fg+fsL (where T.sub.fg+fs=T.sub.fg+T.sub.fs) represents a caustic component, which is a ray that has been reflected off a glossy or specular surface (reference the T.sub.fg+fs operator) before hitting a diffuse surface(reference the T.sub.fd operator); this contribution can be approximated by a high resolution caustic photon map; and
(vi) T.sub.fdT.sub.fdL represents ambient light, which is very smooth and is therefore approximated using a low resolution global photon map.
As noted above, the value integral (equation (6)) is approximated by solving equation (7) making use of sample points x.sub.i, t.sub.i,j and y.sub.i,j,k, where "x.sub.i" refers to sample points within area A.sub.L of the respective pixel atlocation (m,n) in the image plane, "t.sub.i,j" refers to sample points within the time interval t.sub.0 to t.sub.0+T during which the shutter is open, and "y.sub.i,j,k" refers to sample points on the lens A.sub.L. In accordance with one aspect of theinvention, the sample points x.sub.i comprise two-dimensional Hammersley points, which are defined as
.PHI..function. ##EQU00008## where 0.ltoreq.i<N:=(2.sup.n).sup.2 for a selected integer parameter "n," and .PHI..sub.2(i) refers to the radical inverse of "i" in base "two." Generally, the "s" dimensional Hammersley point set is defined as
.times..fwdarw..times..times. .PHI..function..times..PHI..function. ##EQU00009## where I.sup.s is the s-dimensional unit cube [0,1) (that is, an s-dimensional cube that includes "zero," and excludes "one"), the number of points "N" in the setis fixed and b.sub.1, . . . , b.sub.s-1 are bases. The bases do not need to be prime numbers, but they are preferably relatively prime to provide a relatively uniform distribution. The radical inverse function .PHI..sub.b, in turn, is generallydefined as
.PHI..fwdarw..infin..times..times..function..times. .infin..times..times..function..times. ##EQU00010## where (.alpha..sub.j).sub.j=0.sup..infin. is the representation of "i" in integer base "b." The two-dimensional Hammersley points form astratified pattern on a 2.sup.n by 2.sup.n grid. Considering the grid as subpixels, the complete subpixel grid underlying the image plane can be filled by simply abutting copies of the grid to each other.
Given integer subpixel coordinates (s.sub.x,s.sub.y) the instance "i" and coordinates (x,y) for the sample point x.sub.i in the image plane can be determined as follows. Preliminarily, examining
.PHI..function. ##EQU00011## one observes that
(a) each line in the stratified pattern is a shifted copy of another, and
(b) the pattern is symmetric to the line y=x, that is, each column is a shifted copy of another column.
Accordingly, given the integer permutation .sigma.(k):=2.sup.n.PHI..sub.2(k) for 0.ltoreq.k<2.sup.n, subpixel coordinates (s.sub.x,s.sub.y)are mapped onto strata coordinates (j,k):=(s.sub.x mod 2.sup.n,s.sub.y mod 2.sup.n), an instance number"i" is computed as i=j2.sup.n+.sigma.(k) (12) and the positions of the jittered subpixel sample points are determined according to
.PHI..function..PHI..function..sigma..function..sigma..function. ##EQU00012## An efficient algorithm for generating the positions of the jittered subpixel sample points x.sub.i will be provided below in connection with Code Segment 1. A patternof sample points whose positions are determined as described above in connection with equations (12) and (13) has an advantage of having much reduced discrepancy over a pattern determined using a Halton sequence or windowed Halton sequence, as describedin the aforementioned Grabenstein application, and therefore the approximation described above in connection with equation (7) gives in general a better estimation to the value of the integral described above in connection with equation (6). Inaddition, if "n" is sufficiently large, sample points in adjacent pixels will have different patterns, reducing the likelihood that undesirable artifacts will be generated in the image.
A ray tree is a collection of paths of light rays that are traced from a point on the simulated camera's image plane into the scene. The computer graphics system 10 generates a ray tree by recursively following transmission, subsequentreflection and shadow rays using trajectory splitting. In accordance with another aspect of the invention, a path is determined by the components of one vector of a global generalized scrambled Hammersley point set. Generally, a scrambled Hammersleypoint set reduces or eliminates a problem that can arise in connection with higher-dimensioned low-discrepancy sequences since the radical inverse function .PHI..sub.b typically has subsequences of b-1 equidistant values spaced by 1/b . Although thesecorrelation patterns are merely noticeable in the full s-dimensional space, they are undesirable since they are prone to aliasing. The computer graphics system 10 attenuates this effect by scrambling, which corresponds to application of a permutation tothe digits of the b-ary representation used in the radical inversion. For a permutation .sigma. from a symmetric group S.sub.b over integers 0, . . . , b-1, the scrambled radical inverse is defined as
.PHI..times..fwdarw..sigma. .infin..times..sigma..function..function..times..times..revreaction..infi- n..times..function..times..times. ##EQU00013## If the permutation ".sigma." is the identity, the scrambled radical inverse corresponds to theunscrambled radical inverse. In one embodiment, the computer graphics system generates the permutation .sigma. recursively as follows. Starting from the permutation .sigma..sub.2=(0,2) for base b=2, the sequence of permutations is defined as follows:(i) if the base "b" is even, the permutation .sigma..sub.b is generated by first taking the values of
.times..times..sigma. ##EQU00014## and appending the values of
.times..times..sigma. ##EQU00015## and (ii) if the base "b" is odd, the permutation .sigma..sub.b is generated by taking the values of .sigma..sub.b-1, incrementing each value that is greater than or equal to
##EQU00016## by one, and inserting the value b-1 in the middle. This recursive procedure results in
.sigma..sub.2=(0,1)
.sigma..sub.3=(0,1,2)
.sigma..sub.4=(0,2,1,3)
.sigma..sub.5=(0,3,2,1,4)
.sigma..sub.6=(0,2,4,1,3,5)
.sigma..sub.7=(0,2,5,3,1,4,6)
.SIGMA..sub.8=(0,4,2,6,1,5,3,7) . . . .
The computer graphics system 10 can generate a generalized low-discrepancy point set as follows. It is possible to obtain a low-discrepancy sequence by taking any rational s-dimensional point "x" as a starting point and determine a successor byapplying the corresponding incremental radical inverse function to the components of "x." The result is referred to as the generalized low-discrepancy point set. This can be applied to both the Halton sequence and the Hammersley sequence. In the caseof the generalized Halton sequence, this can be formalized as x.sub.i=(.PHI..sub.b.sub.1(i+i.sub.1), .PHI..sub.b.sub.2(i+i.sub.2), . . . , .PHI.(i+i.sub.s)) (15), where the integer vector (i.sub.1, i.sub.2, . . . , i.sub.s) represents the offsets percomponent and is fixed in advance for a generalized sequence. The integer vector can be determined by applying the inverse of the radical inversion to the starting point "x." A generalized Hammersley sequence can be generated in an analogous manner.
Returning to trajectory splitting, generally trajectory splitting is the evaluation of a local integral, which is of small dimension and which makes the actual integrand smoother, which improves overall convergence. Applying replication,positions of low-discrepancy sample points are determined that can be used in evaluating the local integral. The low-discrepancy sample points are shifted by the corresponding elements of the global scrambled Hammersley point set. Since trajectorysplitting can occur multiple times on the same level in the ray tree, branches of the ray tree are decorrelated in order to avoid artifacts, the decorrelation being accomplished by generalizing the global scrambled Hammersley point set.
An efficient algorithm for generating a ray tree will be provided below in connection with Code Segment 2. Generally, in that algorithm, the instance number "i" of the low-discrepancy vector, as determined above in connection with equation (12),and the number "d" of used components, which corresponds to the current integral dimension, are added to the data structure that is maintained for the respective ray in the ray tree. The ray tree of a subpixel sample is completely specified by theinstance number "i." After the dimension has been set to "two," which determines the component of the global Hammersley point set that is to be used next, the primary ray is cast into the scene to span its ray tree. In determining the deterministicsplitting by the components of low discrepancy sample points, the computer graphics system 10 initially allocates the required number of dimensions ".DELTA.d." For example, in simulating glossy scattering, the required number of dimensions willcorrespond to "two." Thereafter, the computer graphics system 10 generates scattering directions from the offset given by the scrambled radical inverses .PHI..sub.b.sub.d(i,.sigma..sub.b.sub.d), . . . ,.PHI..sub.b.sub.d+.DELTA.d-1(i,.sigma..sub.b.sub.d+.DELTA.d-1), yielding the instances
.PHI..function..sigma..sym..times..PHI..DELTA..times..times..function..sig- ma..DELTA..times..times..sym..PHI..DELTA..times..times..function..sigma..D- ELTA..times..times. ##EQU00017## where ".sym." refers to addition modulo "one." Eachdirection of the "M" replicated rays is determined by y.sub.i,j and enters the next level of the ray tree with d':=d+.DELTA.d as the new integral dimension in order to use the next elements of the low-discrepancy vector, and i'=i+j in order todecorrelate subsequent trajectories. Using an infinite sequence of low-discrepancy sample points, the replication heuristic is turned into an adaptive consistent sampling arrangement. That is, computer graphics system 10 can fix the sampling rate.DELTA.M, compare current and previous estimates every .DELTA.M samples, and, if the estimates differ by less than a predetermined threshold value, terminate sampling. The computer graphics system 10 can, in turn, determine the threshold value, byimportance information, that is, how much the local integral contributes to the global integral.
As noted above, the integral described above in connection with equation (6) is over a finite time period T from t.sub.0 to t.sub.0+T, during which time the shutter of the simulated camera is open. During the time period, if an object in thescene moves, the moving object may preferably be depicted in the image as blurred, with the extent of blurring being a function of the object's motion and the time interval t.sub.0+T. Generally, motion during the time an image is recorded is linearlyapproximated by motion vectors, in which case the integrand (equation (6)) is relatively smooth over the time the shutter is open and is suited for correlated sampling. For a ray instance "i," started at the subpixel position x.sub.i, the offset.PHI..sub.3(i) into the time interval is generated and the N.sub.T-1 subsequent samples
.PHI..function..times..times..times..times..times..times.<< ##EQU00018## that is
.PHI..function..sym. ##EQU00019## Determining sample points in this manner fills the sampling space, resulting in a more rapid convergence to the value of the integral (equation (6)). For subsequent trajectory splitting, rays are decorrelatedby setting the instance i'=i+j.
In addition to determining the position of the jittered subpixel sample point x.sub.i, and adjusting the camera and scene according to the sample point t.sub.i,j for the time, the computer graphics system also simulates depth of field. Insimulating depth of field, the camera to be simulated is assumed to be provided with a lens having known optical characteristics and, using geometrical optics, the subpixel sample point x.sub.i is mapped through the lens to yield a mapped point x.sub.i'. The lens is sampled by mapping the dependent samples
.PHI..function..sigma..sym..PHI..function..sigma..sym..PHI..function. ##EQU00020## onto the lens area A.sub.L using a suitable one of a plurality of known transformations. Thereafter, a ray is shot from the sample point on the lens specified byy.sub.i,j,k through the point x.sub.i' into the scene. The offset (.PHI..sub.5(i+j,.sigma..sub.5), .PHI..sub.7(i+j,.sigma..sub.7)) in equation (18) comprise the next components taken from the generalized scrambled Hammersley point set, which, fortrajectory splitting, is displaced by the elements
.PHI..function. ##EQU00021## of the two-dimensional Hammersley point set. The instance of the ray originating from sample point y.sub.i,j,k is set to "i+j+k" in order to decorrelate further splitting down the ray tree. In equation (18), thescrambled samples (.PHI..sub.5(i+j,.sigma..sub.5), .PHI..sub.7(i+j,.sigma..sub.7)) are used instead of the unscrambled samples of (.PHI..sub.5(i+j), .PHI..sub.7(i+j)) since in bases "five" and "seven" the up to five unscrambled samples will lie on astraight line, which will not be the case for the scrambled samples.
In connection with determination of a value for the direct illumination (T.sub.fr-fsL.sub.e above), direct illumination is represented as an integral over the surface of the scene .differential.V, which integral is decomposed into a sum ofintegrals, each over one of the "L" single area light sources in the scene. The individual integrals in turn are evaluated by dependent sampling, that is
.times..times..intg..differential..times..function..times..function..times- ..function..times.d.times..times..times..intg..times..times..times..functi- on..times..function..times..function..times.d.times..times..apprxeq..times-..times..times..function..times..function..times..function. ##EQU00022## where suppL.sub.e,k refers to the surface of the respective "k-th" light source. In evaluating the estimate of the integral for the "k-th" light source, for the M.sub.k-th queryray, shadow rays determine the fraction of visibility of the area light source, since the pont visibility varies much more than the smooth shadow effect. For each light source, the emission L.sub.e is attenuated by a geometric term G, which includes thevisibility, and the surface properties are given by a bidirectional distribution function f.sub.r-f.sub.s. These integrals are local integrals in the ray tree yielding the value of one node in the ray tree, and can be efficiently evaluated usingdependent sampling. In dependent sampling, the query ray comes with the next free integral dimension "d" and the instance "i," from which the dependent samples are determined in accordance with
.PHI..function..sigma..sym..PHI..function..sigma..sym..PHI..function. ##EQU00023## The offset (.PHI..sub.b.sub.d(i,.sigma..sub.b.sub.d), .PHI..sub.b.sub.d+1(i,.sigma..sub.b.sub.d+1)) again is taken from the corresponding generalized scrambledHammersley point set, which shifts the two-dimensional Hammersley point set
.PHI..function. ##EQU00024## on the light source. Selecting the sample rate M.sub.k=2.sup.n.sup.k as a power of two, the local minima is obtained for the discrepancy of the Hammersley point set that perfectly stratifies the light source. As analternative, the light source can be sampled using an arbitrarily-chosen number M.sub.k of sample points using
.PHI..function..sigma. ##EQU00025## as a replication rule. Due to the implicit stratification of the positions of the sample points as described above, the local convergence will be very smooth.
The glossy contribution T.sub.fe(L-L.sub.e) is determined in a manner similar to that described above in connection with area light sources (equations (19) and (20), except that a model f.sub.g used to simulate glossy scattering will be usedinstead of the bidirectional distribution function f.sub.r. In determining the glossy contribution, two-dimensional Hammersley points are generated for a fixed splitting rate M and shifted modulo "one" by the offset(.PHI..sub.b.sub.d(i,.sigma..sub.b.sub.d), .PHI..sub.b.sub.d+1(i,.sigma..sub.b.sub.d+1)) taken from the current ray tree depth given by the dimension field "d" of the incoming ray. The ray trees spanned into the scattering direction are decorrelated byassigning the instance fields i'=i+j in a manner similar to that done for simulation of motion blur and depth of field, as described above. The estimates generated for all rays are averaged by the splitting rate "M" and propagated up the ray tree.
Volumetric effects are typically provided by performing a line integration along respective rays from their origins to the nearest surface point in the scene. In providing for a volumetric effect, the computer graphics system 10 generates fromthe ray data a corresponding offset .PHI..sub.b.sub.d(i) which it then uses to shift the M equidistant samples on the unit interval seen as a one-dimensional torus. In doing so, the rendering time is reduced in comparison to use of an uncorrelatedjittering methodology.
Global illumination includes a class of optical effects, such as indirect illumination, diffuise and glossy inter-reflections, caustics and color bleeding, that the computer graphics system 10 simulates in generating an image of objects in ascene. Simulation of global illumination typically involves the evaluation of a rendering equation. For the general form of an illustrative rendering equation useful in global illumination simulation, namely: L({right arrow over (x)},{right arrow over(w)})=L.sub.e({right arrow over (x)},{right arrow over (w)})+.intg..sub.S'f({right arrow over (x)},{right arrow over (w)}'.fwdarw.{right arrow over (w)})G({right arrow over (x)},{right arrow over (x)}')V({right arrow over (x)},{right arrow over(x)}')L({right arrow over (x)},{right arrow over (w)}')dA' (21) it is recognized that the light radiated at a particular point {right arrow over (x)} in a scene is generally the sum of two components, namely, the amount of light (if any) that is emittedfrom the point and the amount of light (if any) that originates from all other points and which is reflected or otherwise scattered from the point {right arrow over (x)}. In equation (21), L({right arrow over (x)},{right arrow over (w)}) represents theradiance at the point {right arrow over (x)} in the direction {right arrow over (w)}=(.theta.,.phi.) (where ".theta." represents the angle of direction {right arrow over (w)} relative to a direction orthogonal of the surface of the object in the scenecontaining the point {right arrow over (x)}, and ".phi." represents the angle of the component of direction {right arrow over (w)} in a plane tangential to the point {right arrow over (x)}). Similarly, L({right arrow over (x)}',{right arrow over (w)}')in the integral represents the radiance at the point {right arrow over (x)}' in the direction {right arrow over (w)}'=(.theta.',.phi.') (where ".theta.'" represents the angle of direction {right arrow over (w)}' relative to a direction orthogonal of thesurface of the object in the scene containing the point {right arrow over (x)}', and ".phi.'" represents the angle of the component of direction {right arrow over (w)}' in a plane tangential to the point {right arrow over (x)}'), and represents thelight, if any, that is emitted from point {right arrow over (x)}' which may be reflected or otherwise scattered from point {right arrow over (x)}.
Continuing with equation (21), L.sub.e({right arrow over (x)},{right arrow over (w)}) represents the first component of the sum, namely, the radiance due to emission from the point {right arrow over (x)} in the direction {right arrow over (w)},and the integral over the sphere S' represents the second component, namely, the radiance due to scattering of light at point {right arrow over (x)}. f({right arrow over (x)},{right arrow over (w)}'.fwdarw.{right arrow over (w)}) is a bidirectionalscattering distribution function which describes how much of the light coming from direction {right arrow over (w)}' is reflected, refracted or otherwise scattered in the direction {right arrow over (w)}, and is generally the sum of a diffuse component,a glossy component and a specular component. In equation (21), the function G({right arrow over (x)},{right arrow over (x)}') is a geometric term
.function.>>'.times..times..theta..times..times..times..times..theta- .'>>' ##EQU00026## where .theta. and .theta.' are angles relative to the normals of the respective surfaces at points {right arrow over (x)} and {right arrow over(x)}', respectively. Further in equation (21), V({right arrow over (x)},{right arrow over (x)}') is a visibility function which equals the value one if the point {right arrow over (x)}' is visible from the point {right arrow over (x)} and zero if thepoint {right arrow over (x)}' is not visible from the point {right arrow over (x)}.
The computer graphics system 10 makes use of global illumination specifically in connection with determination of the diffuse component T.sub.fdT.sub.fdL and the caustic component T.sub.fdT.sub.fg+fsL using a photon map technique. Generally, aphoton map is constructed by simulating the emission of photons by light source(s) in the scene and tracing the path of each of the photons. For each simulated photon that strikes a surface of an object in the scene, information concerning the simulatedphoton is stored in a data structure referred to as a photon map, including, for example, the simulated photon's color, position and direction angle. Thereafter a Russian roulette operation is performed to determine the photon's state, that is, whetherthe simulated photon will be deemed to have been absorbed or reflected by the surface. If a simulated photon is deemed to have been reflected by the surface, the simulated photon's direction is determined using, for example, a bidirectional reflectancedistribution function ("BRDF"). If the reflected simulated photon strikes another surface, these operations will be repeated (reference the aforementioned Grabenstein application). The data structure in which information for the various simulatedphotons is stored may have any convenient form; typically k-dimensional trees, for "k" an integer" are used. After the photon map has been generated, it can be used in rendering the respective components of the image.
In generating a photon map, the computer graphics system 10 simulates photons trajectories, thus avoiding the necessity of discretizing the kernel of the underlying integral equation. The interactions of the photons with the scene, as describedabove, are stored and used for density estimation. The computer graphics system 10 makes use of a scrambled Halton sequence, which has better discrepancy properties in higher dimensions than does an unscrambled Halton sequence. The scrambled Haltonsequence also has the benefit, over a random sequence, that the approximation error decreases more smoothly, which will allow for use of an adaptive termination scheme during generation of the estimate of the integral. In addition, since the scrambledHalton sequence is strictly deterministic, generation of estimates can readily be parallelized by assigning certain segments of the low-discrepancy sequence to ones of a plurality of processors, which can operate on portions of the computationindependently and in parallel. Since usually the space in which photons will be shot by selecting directions will be much larger than the area of the light sources from which the photons were initially shot, it is advantageous to make use of componentsof smaller discrepancy, for example, .PHI..sub.2 or .PHI..sub.3 (where, as above, .PHI..sub.b refers to the radical inverse function for base "b"), for angular scattering and components of higher discrepancy, for example, .PHI..sub.5 or .PHI..sub.7 forarea sampling, which will facilitate filling the space more uniformly.
The computer graphics system 10 estimates the radiance from the photons in accordance with
.function..omega..apprxeq..times..di-elect cons..function..times..function..omega..omega..times..PHI. ##EQU00027## where, in equation (23), .PHI..sub.i represents the energy of the respective "i-th" photon, .omega..sub.i is the direction ofincidence of the "i-th photon," B.sub.k(x) represents the set of the "k" nearest photons around the point "x," and "A" represents an area around point "x" that includes the photons in the set B.sub.k(x). The computer graphics system 10 makes use of anunbiased but consistent estimator is used for **, which is determined as follows. Taking the radius r(B.sub.k(x)) of the query ball (**what is this?**) a tangential disk D of radius r(B.sub.k(x)) centered on the point x is divided into M equal-sizedsubdomains D.sub.i, that is
.times..times..times..times..times..noteq..times..times..times..times..not- eq..times..times..times..pi..times..times..function..function. ##EQU00028## The set P={D.sub.i|D.sub.i .andgate.{x.sub.i|.sub.D|i .epsilon.B.sub.k(x)}.noteq.0} (25)contains all the subdomains D.sub.i that contain a point x.sub.i|.sub.D on the disk, which is the position of the "i-th" photon projected onto the plane defined by the disk D along its direction of incidence .omega..sub.i. Preferably, the number M ofsubdomains will be on the order of {square root over (k)} and the angular subdivision will be finer than the radial subdivision in order to capture geometry borders. The actual area A is then determined by
.pi..times..times..function..function..times. ##EQU00029## Determining the actual coverage of the disk D by photons significantly improves the radiance estimate (equation (23)) in corners and on borders, where the area obtained by the standardestimate .pi.r.sup.2(B.sub.k(x)) would be too small, which would be the case at corners, or too large, which would be the case at borders. In order to avoid blurring of sharp contours of caustics and shadows, the computer graphics system 10 sets theradiance estimate L to black if all domains D.sub.i that touch x do not contain any photons.
It will be appreciated that, in regions of high photon density, the "k" nearest photons may lead to a radius r(B.sub.k(x) that is nearly zero, which can cause an over-modulation of the estimate. Over-modulation can be avoided by selecting aminimum radius r.sub.min, which will be used if r(B.sub.k(x)) is less than r.sub.min. In that case, instead of equation (23), the estimate is generated in accordance with
.function..omega..times..di-elect cons..function..times..PHI..times..function..omega..omega. ##EQU00030## assuming each photon is started with 1/N of the total flux .PHI.. The estimator in equation (27) provides an estimate for the mean flux ofthe "k" photons if r(B.sub.k(x))<r.sub.min.
The global photon map is generally rather coarse and, as a result, subpixel samples can result in identical photon map queries. As a result, the direct visualization of the global photon map is blurry and it is advantageous to perform asmoothing operation in connection therewith In performing such an operation, the computer graphics system 10 performs a local pass integration that removes artifacts of the direct visualization. Accordingly, the computer graphics system 10 generates anapproximation for the diffuse illumination term T.sub.fdT.sub.fdL as
.times..times..apprxeq..times..times..times..times..intg..function..times.- .function..times..function..function..omega.>.times..times..times..thet- a..times.d.omega..apprxeq..times..function..times..times..function..functi-on..omega..function..times..times..pi..times..times. ##EQU00031## with the integral over the hemisphere S.sup.2(x) of incident directions aligned by the surface normal in "x" being evaluated using importance sampling. The computer graphics system 10stratifies the sample points on a two-dimensional grid by applying dependent trajectory splitting with the Hammersley sequence and thereafter applies irradiance interpolation. Instead of storing the incident flux .PHI..sub.i of the respective photons,the computer graphics system 10 stores their reflected diffuse power f.sub.d.(x.sub.i).PHI..sub.i with the respective photons in the photon map, which allows for a more exact approximation than can be obtained by only sampling the diffuse BRDF in the hitpoints of the final gather rays. In addition, the BRDF evaluation is needed only once per photon, saving the evaluations during the final gathering. Instead of sampling the full grid, the computer graphics system 10 uses adaptive sampling, in whichrefinement is triggered by contrast, distance traveled by the final gather rays in order to more evenly sample the projected solid angle, and the number of photons that are incident form the portion of the projected hemisphere. The computer graphicssystem fills in positions in the grid that are not sampled by interpolation. The resulting image matrix of the projected hemisphere is median filtered in order to remove weak singularities, after which the approximation is generated. The computergraphics system 10 performs the same operation in connection with hemispherical sky illumination.
The computer graphics system 10 processes final gather rays that strike objects that do not cause caustics, such as plane glass windows, by recursive ray tracing. If the hit point of a final gather ray is closer to its origin than apredetermined threshold, the computer graphics system 10 also performs recursive ray tracing. This reduces the likelihood that blurry artifacts will appear in corners, which might otherwise occur since for close hit points the same photons would becollected, which can indirectly expose the blurry structure of the global photon map.
Generally, photon maps have been taken as a snapshot at one point in time, and thus were unsuitable in connection with rendering of motion blur. Following the observation that averaging the result of a plurality of photon maps is generallysimilar to querying one photon map with the total number of photons from all of the plurality of photon maps, the computer graphics system 10 generates N.sub.T photon maps, where N.sub.T is determined as described above, at points in time
.times. ##EQU00032## for 0.ltoreq.b<N.sub.T. During rendering, the computer graphics system 10 uses the photon map with the smallest time difference |t.sub.i,j-t.sub.b| in connection with rendering for the time sample point t.sub.i,j.
The invention provides a number of advantages. In particular, the invention provides a computer graphics system that makes use of strictly deterministic distributed ray tracing based on low-discrepancy sampling and dependent trajectory splittingin connection with rendering of an image of a scene. Generally, strictly deterministic distributed ray tracing based on low-discrepancy sampling and dependent trajectory splitting is simpler to implement than an implementation based on random orpseudo-random numbers. Due to the properties of the radical inversion function, stratification of sample points is intrinsic and does not need to be considered independently of the generation of the positions of the sample points. In addition, sincethe methodology is strictly deterministic, it can be readily parallelized by dividing the image into a plurality of tasks, which can be executed by a plurality of processors in parallel. There is no need to take a step of ensuring that positions ofsample points are not correlated, which is generally necessary if a methodology based on random or pseudo-random numbers is to be implemented for processing in parallel.
Generally, a computer graphics system that makes use of low-discrepancy sampling in determination of sample points will perform better than a computer graphics system that makes use of random or pseudo-random sampling, but the performance maydegrade to that of a system that makes use of random or pseudo-random sampling in higher dimensions. By providing that the computer graphics system performs dependent splitting by replication, the superior convergence of low-dimensional low-discrepancysampling can be exploited with the effect that the overall integrand becomes smoother, resulting in better convergence than with stratified random or pseudo-random sampling. Since the computer graphics system also makes use of dependent trajectorysampling by means of infinite low discrepancy sequences, consistent adaptive sampling of, for example, light sources, can also be performed.
In addition, it will be appreciated that, although the computer graphics system has been described as making use of sample points generated using generalized scrambled and/or unscrambled Hammersley and Halton sequences, it will be appreciatedthat generally any (t,m,s)-net or (t,s)-sequence can be used.
Code Segment 1
The following is a code fragment in the C++ programming language for generating the positions of the jittered subpixel sample points x.sub.i
TABLE-US-00001 unsigned short Period, *Sigma; void InitSigma(int n) { unsigned short Inverse, Digit, Bits; Period = 1 << n; Sigma = new unsigned short [Period]; for (unsigned short i = 0; i < Period; i++) { Digit = Period Inverse = 0;for (bits = i; bits; bits >>= 1) { Digit >>= 1; if(Bits & 1) inverse += Digit; } Sigma[i] = Inverse; } } void SampleSubpixel(unsigned int *i, double *x, double *y, int s.sub.x, int s.sub.y) { int j = s.sub.x & (Period - 1); int k = s.sub.y &(Period - 1); *i = j * Period + Sigma[k] *x = (double) s.sub.x + (double) Sigma[k] / (double) Period; *y = (double) s.sub.y + (double) Sigma[j] / (double) Period; }
Code Segment 2
The following is a code fragment in the C++ programming language a ray tree
TABLE-US-00002 class Ray { int i; //current instance of low discrepancy vector int d; //current integral dimension in ray tree . . . } void Shade (Ray& Ray) { Ray next_ray; int i = ray.i; int d = ray.d . . . for (int j = 0; j < M; j++) { . . . // ray set up for recursion .PHI..function..sigma..sym..PHI..DELTA..function..sigma..DELTA..sym..PHI.- .DELTA..function..sigma..DELTA. ##EQU00033## next_ray = SetUpRay(y); // determine ray parameters by y next_ray.i = i + j; // decorrelation bygeneralization next_ray.d = d + .DELTA.d; //dimension allocation Shade(next_ray); . . . } }
It will be appreciated that a system in accordance with the invention can be constructed in whole or in part from special purpose hardware or a general purpose computer system, or any combination thereof, any portion of which may be controlled bya suitable program. Any program may in whole or in part comprise part of or be stored on the system in a conventional manner, or it may in whole or in part be provided in to the system over a network or other mechanism for transferring information in aconventional manner. In addition, it will be appreciated that the system may be operated and/or otherwise controlled by means of information provided by an operator using operator input elements (not shown) which may be connected directly to the systemor which may transfer the information to the system over a network or other mechanism for transferring information in a conventional manner.
The foregoing description has been limited to a specific embodiment of this invention. It will be apparent, however, that various variations and modifications may be made to the invention, with the attainment of some or all of the advantages ofthe invention. It is the object of the appended claims to cover these and such other variations and modifications as come within the true spirit and scope of the invention.
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